Recent Trends in Combinatorics

Section 1: Extremal and Probabilistic Combinatorics -- Problems Related to Graph Indices in Trees -- The edit distance in graphs: methods, results and generalizations -- Repetitions in graphs and sequences -- On Some Extremal Problems for Cycles in Graphs -- A survey of Turan problems for expansions -- Survey on matching, packing and Hamilton cycle problems on hypergraphs -- Rainbow Hamilton cycles in random graphs and hypergraphs -- Further applications of the Container Method -- Independent transversals and hypergraph matchings - an elementary approach -- Giant components in random graphs -- Infinite random graphs and properties of metrics -- Nordhaus-Gaddum Problems for Colin de Verdière Type Parameters, Variants of Tree-width, and Related Parameters -- Algebraic aspects of the normalized Laplacian -- Poset-free Families of Subsets.- Mathematics of causal sets -- Section 2: Additive and Analytic Combinatorics -- Lectures on Approximate groups and Hilbert\u27s 5th Problem -- Character sums and arithmetic combinatorics -- On sum-product problem -- Ajtai-Szemerédi Theorems over quasirandom groups -- Section 3: Enumerative and Geometric Combinatorics -- Moments of orthogonal polynomials and combinatorics -- The combinatorics of knot invariants arising from the study of Macdonald polynomials -- Some algorithmic applications of partition functions in combinatorics -- Partition Analysis, Modular Functions, and Computer Algebra -- A survey of consecutive patterns in permutations -- Unimodality Problems in Ehrhart Theory -- Face enumeration on simplicial complexes -- Simplicial and Cellular Trees -- Parametric Polyhedra with at least k Lattice Points: Their Semigroup Structure and the k-Frobenius Problem.- Dynamical Algebraic Combinatorics and the Homomesy Phenomenon. This volume presents some of the research topics discussed at the 2014-2015 Annual Thematic Program Discrete Structures: Analysis and Applications at the Institute for Mathematics and its Applications during Fall 2014, when combinatorics was the focus. Leading experts have written surveys of research problems, making state of the art results more conveniently and widely available. The three-part structure of the volume reflects the three workshops held during Fall 2014. In the first part, topics on extremal and probabilistic combinatorics are presented; part two focuses on additive and analytic combinatorics; and part three presents topics in geometric and enumerative combinatorics. This book will be of use to those who research combinatorics directly or apply combinatorial methods to other fields.https://lib.dr.iastate.edu/math_books/1000/thumbnail.jp

Machine olfaction using time scattering of sensor multiresolution graphs

In this paper we construct a learning architecture for high dimensional time series sampled by sensor arrangements. Using a redundant wavelet decomposition on a graph constructed over the sensor locations, our algorithm is able to construct discriminative features that exploit the mutual information between the sensors. The algorithm then applies scattering networks to the time series graphs to create the feature space. We demonstrate our method on a machine olfaction problem, where one needs to classify the gas type and the location where it originates from data sampled by an array of sensors. Our experimental results clearly demonstrate that our method outperforms classical machine learning techniques used in previous studies

Learning mutational graphs of individual tumour evolution from single-cell and multi-region sequencing data

Background. A large number of algorithms is being developed to reconstruct evolutionary models of individual tumours from genome sequencing data. Most methods can analyze multiple samples collected either through bulk multi-region sequencing experiments or the sequencing of individual cancer cells. However, rarely the same method can support both data types. Results. We introduce TRaIT, a computational framework to infer mutational graphs that model the accumulation of multiple types of somatic alterations driving tumour evolution. Compared to other tools, TRaIT supports multi-region and single-cell sequencing data within the same statistical framework, and delivers expressive models that capture many complex evolutionary phenomena. TRaIT improves accuracy, robustness to data-specific errors and computational complexity compared to competing methods. Conclusions. We show that the application of TRaIT to single-cell and multi-region cancer datasets can produce accurate and reliable models of single-tumour evolution, quantify the extent of intra-tumour heterogeneity and generate new testable experimental hypotheses

Enumeration and Random Generation of Unlabeled Classes of Graphs: A Practical Study of Cycle Pointing and the Dissymmetry Theorem

Our work studies the enumeration and random generation of unlabeled combinatorial classes of unrooted graphs. While the technique of vertex pointing provides a straightforward procedure for analyzing a labeled class of unrooted graphs by first studying its rooted counterpart, the existence of nontrivial symmetries in the unlabeled case causes this technique to break down. Instead, techniques such as the dissymmetry theorem (of Otter) and cycle pointing (of Bodirsky et al.) have emerged in the unlabeled case, with the former providing an enumeration of the class and the latter providing both an enumeration and an unbiased sampler. In this work, we extend the power of the dissymmetry theorem by showing that it in fact provides a Boltzmann sampler for the class in question. We then present an exposition of the cycle pointing technique, with a focus on the enumeration and random generation of the underlying unpointed class. Finally, we apply cycle pointing to enumerate and implement samplers for the classes of distance-hereditary graphs and three-leaf power graphs.Comment: 59 pages, 43 figures. Master's thesis, supervised by J\'er\'emie Lumbroso and Robert Sedgewick. Full code available at https://github.com/alexiriza/unlabeled-graph-sampler

Predicting the direction of stock market prices using random forest

Predicting trends in stock market prices has been an area of interest for researchers for many years due to its complex and dynamic nature. Intrinsic volatility in stock market across the globe makes the task of prediction challenging. Forecasting and diffusion modeling, although effective can't be the panacea to the diverse range of problems encountered in prediction, short-term or otherwise. Market risk, strongly correlated with forecasting errors, needs to be minimized to ensure minimal risk in investment. The authors propose to minimize forecasting error by treating the forecasting problem as a classification problem, a popular suite of algorithms in Machine learning. In this paper, we propose a novel way to minimize the risk of investment in stock market by predicting the returns of a stock using a class of powerful machine learning algorithms known as ensemble learning. Some of the technical indicators such as Relative Strength Index (RSI), stochastic oscillator etc are used as inputs to train our model. The learning model used is an ensemble of multiple decision trees. The algorithm is shown to outperform existing algo- rithms found in the literature. Out of Bag (OOB) error estimates have been found to be encouraging. Key Words: Random Forest Classifier, stock price forecasting, Exponential smoothing, feature extraction, OOB error and convergence

forgeNet: A graph deep neural network model using tree-based ensemble classifiers for feature extraction

A unique challenge in predictive model building for omics data has been the small number of samples $(n)$ versus the large amount of features $(p)$. This "$n\ll p$" property brings difficulties for disease outcome classification using deep learning techniques. Sparse learning by incorporating external gene network information such as the graph-embedded deep feedforward network (GEDFN) model has been a solution to this issue. However, such methods require an existing feature graph, and potential mis-specification of the feature graph can be harmful on classification and feature selection. To address this limitation and develop a robust classification model without relying on external knowledge, we propose a \underline{for}est \underline{g}raph-\underline{e}mbedded deep feedforward \underline{net}work (forgeNet) model, to integrate the GEDFN architecture with a forest feature graph extractor, so that the feature graph can be learned in a supervised manner and specifically constructed for a given prediction task. To validate the method's capability, we experimented the forgeNet model with both synthetic and real datasets. The resulting high classification accuracy suggests that the method is a valuable addition to sparse deep learning models for omics data

Periodic Jacobi Matrices on Trees

We begin the systematic study of the spectral theory of periodic Jacobi matrices on trees including a formal definition. The most significant result that appears here for the first time is that these operators have no singular continuous spectrum. We review important previous results of Sunada and Aomoto and present several illuminating examples. We present many open problems and conjectures that we hope will stimulate further work.Comment: appeared as Adv. Math. 379 (2020), 107241; this version includes an addendu

Spanning trees in random series-parallel graphs

By means of analytic techniques we show that the expected number of spanning trees in a connected labelled series-parallel graph on $n$ vertices chosen uniformly at random satisfies an estimate of the form $s \varrho^{-n} (1+o(1))$, where $s$ and $\varrho$ are computable constants, the values of which are approximately $s \approx 0.09063$ and $\varrho^{-1} \approx 2.08415$. We obtain analogue results for subfamilies of series-parallel graphs including 2-connected series-parallel graphs, 2-trees, and series-parallel graphs with fixed excess

MAP estimation via agreement on (hyper)trees: Message-passing and linear programming

We develop and analyze methods for computing provably optimal {\em maximum a posteriori} (MAP) configurations for a subclass of Markov random fields defined on graphs with cycles. By decomposing the original distribution into a convex combination of tree-structured distributions, we obtain an upper bound on the optimal value of the original problem (i.e., the log probability of the MAP assignment) in terms of the combined optimal values of the tree problems. We prove that this upper bound is tight if and only if all the tree distributions share an optimal configuration in common. An important implication is that any such shared configuration must also be a MAP configuration for the original distribution. Next we develop two approaches to attempting to obtain tight upper bounds: (a) a {\em tree-relaxed linear program} (LP), which is derived from the Lagrangian dual of the upper bounds; and (b) a {\em tree-reweighted max-product message-passing algorithm} that is related to but distinct from the max-product algorithm. In this way, we establish a connection between a certain LP relaxation of the mode-finding problem, and a reweighted form of the max-product (min-sum) message-passing algorithm.Comment: Presented in part at the Allerton Conference on Communication, Computing and Control in October 2002. Full journal version appear in the IEEE Transactions on Information Theory, November 200

Blind prediction of protein B-factor and flexibility

Debye-Waller factor, a measure of X-ray attenuation, can be experimentally observed in protein X-ray crystallography. Previous theoretical models have made strong inroads in the analysis of B-factors by linearly fitting protein B-factors from experimental data. However, the blind prediction of B-factors for unknown proteins is an unsolved problem. This work integrates machine learning and advanced graph theory, namely, multiscale weighted colored graphs (MWCGs), to blindly predict B-factors of unknown proteins. MWCGs are local features that measure the intrinsic flexibility due to a protein structure. Global features that connect the B-factors of different proteins, e.g., the resolution of X-ray crystallography, are introduced to enable the cross-protein B-factor predictions. Several machine learning approaches, including ensemble methods and deep learning, are considered in the present work. The proposed method is validated with hundreds of thousands of experimental B-factors. Extensive numerical results indicate that the blind B-factor predictions obtained from the present method are more accurate than the least squares fittings using traditional methods.Comment: 5 figures, 23 page